Extremal spectral properties of Lawson tau-surfaces and the Lam\'e equation

TL;DR

This paper investigates the extremal spectral properties of Lawson tau-surfaces, explicitly determining extremal eigenvalues of the Laplace-Beltrami operator using Lamé equation theory, contributing to the understanding of minimal surfaces in spheres.

Contribution

It explicitly computes extremal eigenvalues of Lawson tau-surfaces by applying Lamé equation theory, revealing new spectral extremal properties of these minimal surfaces.

Findings

Explicit extremal eigenvalues for Lawson tau-surfaces

Connection between Lamé equation and spectral properties

Identification of extremal metrics on minimal surfaces

Abstract

Extremal spectral properties of Lawson tau-surfaces are investigated. The Lawson tau-surfaces form a two-parametric family of tori or Klein bottles minimally immersed in the standard unitary three-dimensional sphere. A Lawson tau-surface carries an extremal metric for some eigenvalue of the Laplace-Beltrami operator. Using theory of the Lam\'e equation we find explicitly these extremal eigenvalues.